I think you're confusing "represented" and "located". Yes, all real numbers can be represented by the points on a line. But remember that a line is an abstract thing. You can never draw or imagine a line (only represent it approximately). Once you realise these things you should be more clear about this (admittedly deep matter -- check out "continuum").
So, yes, whereas every point on a straight line can be made to correspond to one and only one real number, it does not mean we can always locate this point for all real numbers (by locate I mean using a finite sequence of steps). You may call this theology, but math sometimes often deals in such things, especially in relation to deep matters like this.
The real numbers we can locate are known as constructible numbers. But perhaps even if we allow ourselves more freedom, we may even find rectilinear lengths of any real size and use it to "locate" our points.
But don't think all is settled, or that you can finish settling this in a few hours. It's a very interesting matter that one keeps getting back to.
So how should one think rigorously of real numbers (wlog irrational numbers)? Well, they were defined because extracting the roots and logarithms of rationals does not always give a rational value. But since we know these things must correspond to some quantity, we call them irrationals (this is not very different from the invention of negative numbers and complex numbers). The special thing about irrational numbers is that the involve the "cumulation" of infinitely many operations, loosely speaking. Dedekind was first to give a rigorous definition in terms of order relations and sets (so-called Dedekind cuts).